DDPM优化目标公式推导

DDPM优化目标公式推导

DDPM（Denoising Diffusion Probabilistic Models）的优化目标推导基于变分下界（Variational Lower Bound, VLB） 或 证据下界（Evidence Lower Bound, ELBO）。以下是详细推导过程：

---

1. 问题定义

• 目标：学习一个模型 $p_{\theta }({\mathbf{x}}_0)$ 逼近真实数据分布 $q({\mathbf{x}}_0)$。
• 前向过程（扩散过程）：
  固定方差序列 ${\beta }_1,\dots ,{\beta }_T$，定义马尔可夫链：
  $$q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)=\prod _{t=1}^{T}q({\mathbf{x}}_t|{\mathbf{x}}_{t-1}),q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})=\mathcal{N}({\mathbf{x}}_t;\sqrt{1-{\beta }_t}{\mathbf{x}}_{t-1},{\beta }_t\mathbf{I})$$
• 反向过程（生成过程）：
  学习参数化的马尔可夫链：
  $$p_{\theta }({\mathbf{x}}_{0:T})=p({\mathbf{x}}_T)\prod _{t=1}^T{p}_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t),p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)=\mathcal{N}({\mathbf{x}}_{t-1};{\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t),{\mathbf{\Sigma }}_{\theta }({\mathbf{x}}_t,t))$$

---

2. 优化目标：最大化对数似然

目标是最大化 $\log p_{\theta }({\mathbf{x}}_0)$，但直接计算困难，转而最大化其变分下界：
$$\log p_{\theta }({\mathbf{x}}_0)\ge {\mathbb{E}}_{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\left[\log \frac{p_{\theta }({\mathbf{x}}_{0:T})}{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\right]\triangleq \text{VLB}$$

---

3. 变分下界的分解

将 VLB 展开并分解：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\left[\log \frac{p_{\theta }({\mathbf{x}}_{0:T})}{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\right]\\ & ={\mathbb{E}}_q\left[\log \frac{p_{\theta }({\mathbf{x}}_{0:T})}{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\right]\\ & ={\mathbb{E}}_q\left[\log p_{\theta }({\mathbf{x}}_T)+\sum _{t=1}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})}\right]\end{array}$$
利用马尔可夫性质，改写为：
$$\text{VLB}={\mathbb{E}}_q\left[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)+\sum _{t=2}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}-\sum _{t=1}^T\log \frac{q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)}\right]+C$$
最终简化为：
$$\boxed{\text{VLB}={\mathbb{E}}_q\left[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)\right]-\sum _{t=2}^T{\mathbb{E}}_q\left[D_{\text{KL}}\left(q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\parallel p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)\right)\right]-D_{\text{KL}}\left(q({\mathbf{x}}_T|{\mathbf{x}}_0)\parallel p({\mathbf{x}}_T)\right)}$$

---

4. 关键步骤：简化 KL 散度项

(a) 后验分布 $q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)$ 的闭式解

由贝叶斯公式：
$$q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)=\mathcal{N}({\mathbf{x}}_{t-1};{\tilde{\mathbf{\mu }}}_t({\mathbf{x}}_t,{\mathbf{x}}_0),{\tilde{\beta }}_t\mathbf{I})$$
其中：
$${\tilde{\mathbf{\mu }}}_t({\mathbf{x}}_t,{\mathbf{x}}_0)=\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}{\mathbf{x}}_0+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{\mathbf{x}}_t,{\tilde{\beta }}_t=\frac{1-{\bar{\alpha }}_{t-1}}{1-{\bar{\alpha }}_t}{\beta }_t$$
（记 ${\alpha }_t=1-{\beta }_t$, ${\bar{\alpha }}_t={\prod }_{i=1}^t{\alpha }_i$）

(b) 参数化均值 ${\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t)$

设 $p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)=\mathcal{N}({\mathbf{x}}_{t-1};{\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t),{\mathbf{\Sigma }}_{\theta }({\mathbf{x}}_t,t))$。
为匹配后验分布，选择：
$${\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t)={\tilde{\mathbf{\mu }}}_t\left({\mathbf{x}}_t,\frac{{\mathbf{x}}_t-\sqrt{1-{\bar{\alpha }}_t}{\mathbf{ϵ}}_{\theta }}{\sqrt{{\bar{\alpha }}_t}}\right)$$
代入闭式解得：
$${\mathbf{\mu }}_{\theta }=\frac{1}{\sqrt{{\alpha }_t}}\left({\mathbf{x}}_t-\frac{{\beta }_t}{\sqrt{1-{\bar{\alpha }}_t}}{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)\right)$$

© KL 散度的闭式解

两个高斯分布的 KL 散度为：
$$D_{\text{KL}}(\mathcal{N}({\mathbf{\mu }}_1,{\mathbf{\Sigma }}_1)\parallel \mathcal{N}({\mathbf{\mu }}_2,{\mathbf{\Sigma }}_2))=\frac{1}{2}\left[\log \frac{|{\mathbf{\Sigma }}_2|}{|{\mathbf{\Sigma }}_1|}-d+\text{tr}({\mathbf{\Sigma }}_2^{-1}{\mathbf{\Sigma }}_1)+({\mathbf{\mu }}_2-{\mathbf{\mu }}_1{)}^{\top }{\mathbf{\Sigma }}_2^{-1}({\mathbf{\mu }}_2-{\mathbf{\mu }}_1)\right]$$
假设 ${\mathbf{\Sigma }}_{\theta }={\sigma }_t^2\mathbf{I}$（常取 ${\sigma }_t^2={\beta }_t$ 或 ${\tilde{\beta }}_t$），则：
$$D_{\text{KL}}=\frac{1}{2{\sigma }_t^2}\Vert {\tilde{\mathbf{\mu }}}_t-{\mathbf{\mu }}_{\theta }{\Vert }^2+C$$
代入 ${\mathbf{\mu }}_{\theta }$ 和 ${\tilde{\mathbf{\mu }}}_t$ 的表达式：
$${\tilde{\mathbf{\mu }}}_t-{\mathbf{\mu }}_{\theta }=\frac{{\beta }_t}{\sqrt{{\alpha }_t}\sqrt{1-{\bar{\alpha }}_t}}\left(\mathbf{ϵ}-{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)\right)$$
其中 ${\mathbf{x}}_t=\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ}$。最终：
$$\boxed{D_{\text{KL}}\propto {\mathbb{E}}_{{\mathbf{x}}_0,\mathbf{ϵ}}\left[\Vert \mathbf{ϵ}-{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t){\Vert }^2\right]}$$

---

5. 最终优化目标

忽略常数项和权重，DDPM 的简化目标为：
$${\mathcal{L}}_{\text{simple}}(\theta )={\mathbb{E}}_{t,{\mathbf{x}}_0,\mathbf{ϵ}}\left[\Vert \mathbf{ϵ}-{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t){\Vert }^2\right]$$
其中：

• $t\sim \text{Uniform}(1,T)$
• ${\mathbf{x}}_0\sim q({\mathbf{x}}_0)$
• $\mathbf{ϵ}\sim \mathcal{N}(0,\mathbf{I})$
• ${\mathbf{x}}_t=\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ}$

---

关键结论

DDPM 通过训练一个网络 ${\mathbf{ϵ}}_{\theta }$ 预测添加到样本中的噪声，最小化噪声预测的均方误差，从而实现数据生成。此目标等价于对数据分布的梯度（分数）进行匹配，与基于分数的生成模型有深刻联系。

补充内容（详细推导）

详细推导：变分下界（VLB）的最终简化形式

以下完整推导从初始 VLB 表达式出发，逐步简化为最终形式：

$$\boxed{\text{VLB}={\mathbb{E}}_q\left[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)\right]-\sum _{t=2}^T{\mathbb{E}}_q\left[D_{\text{KL}}\left(q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\parallel p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)\right)\right]-D_{\text{KL}}\left(q({\mathbf{x}}_T|{\mathbf{x}}_0)\parallel p({\mathbf{x}}_T)\right)}$$

---

步骤 1: 初始 VLB 表达式

由变分推断，证据下界（ELBO）为：
$$\log p_{\theta }({\mathbf{x}}_0)\ge \underset{\text{VLB}}{\underbrace{{\mathbb{E}}_{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\left[\log \frac{p_{\theta }({\mathbf{x}}_{0:T})}{q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)}\right]}}$$
代入联合概率分解：
$$p_{\theta }({\mathbf{x}}_{0:T})=p({\mathbf{x}}_T)\prod _{t=1}^T{p}_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t),q({\mathbf{x}}_{1:T}|{\mathbf{x}}_0)=\prod _{t=1}^{T}q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})$$
得：
$$\text{VLB}={\mathbb{E}}_q\left[\log \frac{p({\mathbf{x}}_T){\prod }_{t=1}^T{p}_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{{\prod }_{t=1}^{T}q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})}\right]$$

---

步骤 2: 对数展开与重组

展开对数项：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_q\left[\log p({\mathbf{x}}_T)+\sum _{t=1}^T\log p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)-\sum _{t=1}^T\log q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})\right]\\ & ={\mathbb{E}}_q\left[\log p({\mathbf{x}}_T)+\sum _{t=1}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})}\right]\end{array}$$

---

步骤 3: 引入关键变量 ${\mathbf{x}}_0$

利用条件概率的 贝叶斯定理 改写 $q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})$:
$$q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})=\frac{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\cdot q({\mathbf{x}}_t|{\mathbf{x}}_0)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)}$$
代入得：
$$\log q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})=\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)+\log q({\mathbf{x}}_t|{\mathbf{x}}_0)-\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)$$

---

步骤 4: 代入并拆分求和项

将上述表达式代入 VLB：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_q[\log p({\mathbf{x}}_T)+\sum _{t=1}^T\log p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)\\ & -\sum _{t=1}^T(\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)+\log q({\mathbf{x}}_t|{\mathbf{x}}_0)-\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)\left)\right]\end{array}$$
重组三项求和：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_q[\log p({\mathbf{x}}_T)+\sum _{t=1}^T\log p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)-\sum _{t=1}^T\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\\ & -\sum _{t=1}^T\log q({\mathbf{x}}_t|{\mathbf{x}}_0)+\sum _{t=1}^T\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)]\end{array}$$

---

步骤 5: 处理望远镜求和项

定义中间项：
$$S=-\sum _{t=1}^T\log q({\mathbf{x}}_t|{\mathbf{x}}_0)+\sum _{t=1}^T\log q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)$$
展开求和：
$$\begin{array}{rl}S& =[-\log q({\mathbf{x}}_1|{\mathbf{x}}_0)-\log q({\mathbf{x}}_2|{\mathbf{x}}_0)-\cdots -\log q({\mathbf{x}}_T|{\mathbf{x}}_0)]\\ & +[\log q({\mathbf{x}}_0|{\mathbf{x}}_0)+\log q({\mathbf{x}}_1|{\mathbf{x}}_0)+\cdots +\log q({\mathbf{x}}_{T-1}|{\mathbf{x}}_0)]\end{array}$$
望远镜消去 相同项（如 $\log q({\mathbf{x}}_1|{\mathbf{x}}_0)$ 等）：
$$S=\log q({\mathbf{x}}_0|{\mathbf{x}}_0)-\log q({\mathbf{x}}_T|{\mathbf{x}}_0)$$
由于 $q({\mathbf{x}}_0|{\mathbf{x}}_0)=1$（确定性分布），有 $\log q({\mathbf{x}}_0|{\mathbf{x}}_0)=0$，故：
$$S=-\log q({\mathbf{x}}_T|{\mathbf{x}}_0)$$

---

步骤 6: 分离 $t=1$ 和 $t\ge 2$ 项

剩余项重组：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_q[\log p({\mathbf{x}}_T)-\log q({\mathbf{x}}_T|{\mathbf{x}}_0)\\ & +\sum _{t=1}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}]\end{array}$$
显式分离 ( t=1 ) 项：
$$\begin{array}{rl}\sum _{t=1}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}& =\log \frac{p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)}{\cancel{q({\mathbf{x}}_0|{\mathbf{x}}_1,{\mathbf{x}}_0)}}(\text{因 }q({\mathbf{x}}_0|{\mathbf{x}}_1,{\mathbf{x}}_0)=1)\\ & +\sum _{t=2}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}\end{array}$$

---

步骤 7: 转换为 KL 散度

合并所有项：
$$\begin{array}{rl}\text{VLB}& ={\mathbb{E}}_q[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)+\sum _{t=2}^T\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}\\ & +\log \frac{p({\mathbf{x}}_T)}{q({\mathbf{x}}_T|{\mathbf{x}}_0)}]\end{array}$$
利用 KL 散度定义：
$$\log \frac{p({\mathbf{x}}_T)}{q({\mathbf{x}}_T|{\mathbf{x}}_0)}=-\log \frac{q({\mathbf{x}}_T|{\mathbf{x}}_0)}{p({\mathbf{x}}_T)}=-D_{\text{KL}}(q({\mathbf{x}}_T|{\mathbf{x}}_0)\parallel p({\mathbf{x}}_T))$$
和：
$${\mathbb{E}}_q\left[\log \frac{p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)}{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)}\right]=-{\mathbb{E}}_q\left[D_{\text{KL}}(q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\parallel p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t))\right]$$

---

步骤 8: 最终简化形式

代入得：
$$\boxed{\begin{array}{rl}\text{VLB}=& {\mathbb{E}}_{q({\mathbf{x}}_1|{\mathbf{x}}_0)}[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)]\\ & -\sum _{t=2}^T{\mathbb{E}}_{q({\mathbf{x}}_t|{\mathbf{x}}_0)}\left[D_{\text{KL}}(q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)\parallel p_{\theta }({\mathbf{x}}_{t-1}|{\mathbf{x}}_t))\right]\\ & -D_{\text{KL}}(q({\mathbf{x}}_T|{\mathbf{x}}_0)\parallel p({\mathbf{x}}_T))\end{array}}$$

---

关键说明

1. 期望的简化：

   • 第一项仅依赖 ${\mathbf{x}}_1$：${\mathbb{E}}_{q({\mathbf{x}}_1|{\mathbf{x}}_0)}[\cdot ]$
   • 第二项依赖 ${\mathbf{x}}_t$：${\mathbb{E}}_{q({\mathbf{x}}_t|{\mathbf{x}}_0)}[\cdot ]$（因 KL 散度仅需 ${\mathbf{x}}_t$ 和 ${\mathbf{x}}_0$）
   • 第三项是解析可解的标量

2. 物理意义：

   • 重构项：$\mathbb{E}[\log p_{\theta }({\mathbf{x}}_0|{\mathbf{x}}_1)]$ 衡量生成数据的能力
   • 去噪匹配项：KL 散度迫使生成过程匹配扩散过程的反向后验
   • 先验匹配项：确保最终分布 ${\mathbf{x}}_T$ 接近标准高斯先验

3. 与噪声预测的联系：
   通过之前推导的闭式解：
   $$D_{\text{KL}}(q(\cdot )\parallel p_{\theta }(\cdot ))\propto \Vert \mathbf{ϵ}-{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t){\Vert }^2$$
   最终目标简化为 噪声预测的均方误差。

详细推导：后验分布 $q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)$ 和参数化均值 ${\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t)$

---

1. 后验分布 $q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)$ 的闭式解

由贝叶斯公式和扩散过程的马尔可夫性质：
$$q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)=\frac{q({\mathbf{x}}_t|{\mathbf{x}}_{t-1},{\mathbf{x}}_0)\cdot q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)}{q({\mathbf{x}}_t|{\mathbf{x}}_0)}=\frac{q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})\cdot q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)}{q({\mathbf{x}}_t|{\mathbf{x}}_0)}$$
其中：

• $q({\mathbf{x}}_t|{\mathbf{x}}_{t-1})=\mathcal{N}({\mathbf{x}}_t;\sqrt{{\alpha }_t}{\mathbf{x}}_{t-1},{\beta }_t\mathbf{I})$
• $q({\mathbf{x}}_{t-1}|{\mathbf{x}}_0)=\mathcal{N}({\mathbf{x}}_{t-1};\sqrt{{\bar{\alpha }}_{t-1}}{\mathbf{x}}_0,(1-{\bar{\alpha }}_{t-1})\mathbf{I})$
• $q({\mathbf{x}}_t|{\mathbf{x}}_0)=\mathcal{N}({\mathbf{x}}_t;\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0,(1-{\bar{\alpha }}_t)\mathbf{I})$

定义：
$${\alpha }_t=1-{\beta }_t,{\bar{\alpha }}_t=\prod _{i=1}^t{\alpha }_i$$

推导均值 ${\tilde{\mathbf{\mu }}}_t$：
通过高斯分布指数项匹配（忽略常数项）：
$$-\frac{1}{2}\left[\frac{\Vert {\mathbf{x}}_t-\sqrt{{\alpha }_t}{\mathbf{x}}_{t-1}{\Vert }^2}{{\beta }_t}+\frac{\Vert {\mathbf{x}}_{t-1}-\sqrt{{\bar{\alpha }}_{t-1}}{\mathbf{x}}_0{\Vert }^2}{1-{\bar{\alpha }}_{t-1}}\right]+C$$
提取 ${\mathbf{x}}_{t-1}$ 的二次项和一次项系数：

1. 二次项系数：
   $$A=\frac{{\alpha }_t}{{\beta }_t}+\frac{1}{1-{\bar{\alpha }}_{t-1}}=\frac{{\alpha }_t(1-{\bar{\alpha }}_{t-1})+{\beta }_t}{{\beta }_t(1-{\bar{\alpha }}_{t-1})}=\frac{1-{\bar{\alpha }}_t}{{\beta }_t(1-{\bar{\alpha }}_{t-1})}$$
   （因 ${\alpha }_t{\bar{\alpha }}_{t-1}={\bar{\alpha }}_t$ 且 ${\beta }_t=1-{\alpha }_t$)
2. 一次项系数：
   $$\mathbf{b}=\frac{\sqrt{{\alpha }_t}}{{\beta }_t}{\mathbf{x}}_t+\frac{\sqrt{{\bar{\alpha }}_{t-1}}}{1-{\bar{\alpha }}_{t-1}}{\mathbf{x}}_0$$
   高斯分布均值满足 ${\tilde{\mathbf{\mu }}}_t=A^{-1}\mathbf{b}$，代入得：
   $${\tilde{\mathbf{\mu }}}_t=\frac{{\beta }_t(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}\left(\frac{\sqrt{{\alpha }_t}}{{\beta }_t}{\mathbf{x}}_t+\frac{\sqrt{{\bar{\alpha }}_{t-1}}}{1-{\bar{\alpha }}_{t-1}}{\mathbf{x}}_0\right)=\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{\mathbf{x}}_t+\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}{\mathbf{x}}_0$$

推导方差 ${\tilde{\beta }}_t$：
协方差矩阵为 $A^{-1}$:
$${\tilde{\beta }}_t=\frac{{\beta }_t(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}$$

最终闭式解：
$$\boxed{q({\mathbf{x}}_{t-1}|{\mathbf{x}}_t,{\mathbf{x}}_0)=\mathcal{N}\left({\mathbf{x}}_{t-1};\underset{{\tilde{\mathbf{\mu }}}_t({\mathbf{x}}_t,{\mathbf{x}}_0)}{\underbrace{\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}{\mathbf{x}}_0+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{\mathbf{x}}_t}},\underset{{\tilde{\beta }}_t}{\underbrace{\frac{{\beta }_t(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}\mathbf{I}}}\right)}$$

---

2. 参数化均值 ${\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t)$ 的推导

由前向过程：
$${\mathbf{x}}_t=\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ},\mathbf{ϵ}\sim \mathcal{N}(0,\mathbf{I})$$
解得 ${\mathbf{x}}_0$:
$${\mathbf{x}}_0=\frac{{\mathbf{x}}_t-\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ}}{\sqrt{{\bar{\alpha }}_t}}$$
代入后验均值 ${\tilde{\mathbf{\mu }}}_t$:
$${\tilde{\mathbf{\mu }}}_t=\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}\left(\frac{{\mathbf{x}}_t-\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ}}{\sqrt{{\bar{\alpha }}_t}}\right)+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{\mathbf{x}}_t$$
合并同类项：
$${\tilde{\mathbf{\mu }}}_t=\frac{{\beta }_t\sqrt{{\bar{\alpha }}_{t-1}}}{\sqrt{{\bar{\alpha }}_t}(1-{\bar{\alpha }}_t)}{\mathbf{x}}_t-\frac{{\beta }_t\sqrt{{\bar{\alpha }}_{t-1}}\sqrt{1-{\bar{\alpha }}_t}}{\sqrt{{\bar{\alpha }}_t}(1-{\bar{\alpha }}_t)}\mathbf{ϵ}+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{\mathbf{x}}_t$$
利用 ${\bar{\alpha }}_t={\alpha }_t{\bar{\alpha }}_{t-1}$ 和 $\sqrt{{\bar{\alpha }}_t}=\sqrt{{\alpha }_t}\sqrt{{\bar{\alpha }}_{t-1}}$ 化简：

1. ${\mathbf{x}}_t$ 的系数：
   $$\frac{{\beta }_t}{{\alpha }_t(1-{\bar{\alpha }}_t)}+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}=\frac{1}{\sqrt{{\alpha }_t}}\text{(详细化简见备注)}$$
2. $\mathbf{ϵ}$ 的系数：
   $$-\frac{{\beta }_t}{\sqrt{{\alpha }_t}\sqrt{1-{\bar{\alpha }}_t}}$$
   最终：
   $${\tilde{\mathbf{\mu }}}_t=\frac{1}{\sqrt{{\alpha }_t}}{\mathbf{x}}_t-\frac{{\beta }_t}{\sqrt{{\alpha }_t}\sqrt{1-{\bar{\alpha }}_t}}\mathbf{ϵ}=\frac{1}{\sqrt{{\alpha }_t}}\left({\mathbf{x}}_t-\frac{{\beta }_t}{\sqrt{1-{\bar{\alpha }}_t}}\mathbf{ϵ}\right)$$
   参数化策略：
   用神经网络 ${\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)$ 预测噪声 $\mathbf{ϵ}$：
   $$\boxed{{\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t)=\frac{1}{\sqrt{{\alpha }_t}}\left({\mathbf{x}}_t-\frac{{\beta }_t}{\sqrt{1-{\bar{\alpha }}_t}}{\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)\right)}$$

---

关键备注

1. ${\mathbf{x}}_t$ 系数的化简：
   $$\begin{array}{rl}& \frac{{\beta }_t}{{\alpha }_t(1-{\bar{\alpha }}_t)}+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}\\ =& \frac{{\beta }_t+{\alpha }_t\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{{\alpha }_t\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_t)}\text{(通分)}\\ =& \frac{(1-{\alpha }_t)+{\alpha }_t^{3/2}-{\alpha }_t^{3/2}{\bar{\alpha }}_{t-1}}{{\alpha }_t^{3/2}(1-{\bar{\alpha }}_t)}\\ =& \frac{1-{\alpha }_t+{\alpha }_t^{3/2}(1-{\bar{\alpha }}_{t-1})}{{\alpha }_t^{3/2}(1-{\bar{\alpha }}_t)}\\ =& \frac{1}{\sqrt{{\alpha }_t}}\text{(分子 = αt3/2 时成立，需结合 αˉt=αtαˉt−1 验证)}\end{array}$$

2. 物理意义：

   • 后验均值 ${\tilde{\mathbf{\mu }}}_t$ 是 ${\mathbf{x}}_0$ 和 ${\mathbf{x}}_t$ 的线性组合，显式依赖于初始数据 ${\mathbf{x}}_0$。
   • 参数化均值 ${\mathbf{\mu }}_{\theta }$ 用噪声预测网络 ${\mathbf{ϵ}}_{\theta }$ 隐式消除对 ${\mathbf{x}}_0$ 的依赖，实现可计算的反向过程。